Example of Exact Trade-Offs in Linear Controller Design

C. Barratt and S. Boyd

IEEE Control Systems Magazine, 9(1):46-52, January 1989.
Conference paper:
Exact tradeoffs in LTI controller design: An example, Proceedings of the
American Control Conference, pages 1274-1279, June 1989.

The design of a linear time-invariant controller for a given
linear-time-invariant plant, like any engineering design, involves tradeoff
among many desirable qualities, such as fast response to commands without
excessive overshoot low actuator authority, robustness low controller
complexity, and so on. Only for a few very special cases are analytic methods
known for finding the exact form of these tradeoffs. Two examples of such
analytic methods are Linear Quadratic Gaussian (LQG) theory (where the plant
actuator and output variance can be traded off), and Nevanlinna-Pick (NP)
theory (where, for instance, the achievable disturbance rejection can be traded
off in two different bandwidths). In many cases the limit of performance
achievable with a linear-time-invariant controller, and thus the exact form of
the tradeoffs, can be computed numerically. To demonstrate this tradeoff, the
paper treats the design of a regulator for a very typical plant, a double
integrator with some excess phase. The tradeoffs are presented for two
different measures of robustness with noise sensitivity. The exact form of
these tradeoffs is numerically determined using the techniques described in the
appendix.